We study stochastic linear--quadratic (LQ) optimal control problems over an infinite horizon, allowing the cost matrices to be indefinite. We develop a systematic approach based on semidefinite programming (SDP). A central issue is the stability of the feedback control; and we show this can be effectively examined through the complementary duality of the SDP. Furthermore, we establish several implication relations among the SDP complementary duality, the (generalized) Riccati equation, and the optimality of the LQ control problem. Based on these relations, we propose a numerical procedure that provides a thorough treatment of the LQ control problem via SDP: it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution. For the latter case, we develop an ε-approximation scheme that is asymptotically optimal.

complementary duality, generalized Riccati equation, mean-square stability, semidefinite programming, stochastic LQ control
hdl.handle.net/1765/1590
Econometric Institute Research Papers
Erasmus School of Economics

Yao, D.D, Zhang, S, & Zhou, X.Y. (1999). LQ Control without Riccati Equations: Stochastic Systems (No. EI 9920-/A). Econometric Institute Research Papers. Retrieved from http://hdl.handle.net/1765/1590